A non-perturbative expansion method which gives a well-defined analyti
c continuation of the running coupling constant from the spacelike to
the timelike region is applied to the inclusive semileptonic decay of
the tau-lepton. The method allows us to evaluate R(tau) by integration
over the non-perturbative physical region, rather than by using Cauch
y's theorem, and hence to incorporate threshold effects in a very dire
ct way. Within our framework the difference between the effective coup
ling constants in the timelike and spacelike domains can be substantia
l and is not simply a matter of the standard pi(2) terms.